It is a common goal of various measurement systems to enhance the resolution of measured data. Such measurement systems include for example optical imaging systems, where the resolution is generally limited by diffraction limit, i.e. defining the smallest resolvable feature in optical imaging of the specific imaging system, which is determined mainly by the numerical aperture of optical components (lenses, etc.) involved. However, even a system with an infinite aperture has a resolution limit, which arises from the wavelength λ of an electromagnetic (EM) field. Therefore, the best recoverable resolution of the optical system is λ/2 regardless. This is because the propagation of EM waves in bulk media acts as a low-pass filter, for distances much larger than the wavelength, rendering spatial frequencies larger than 1/λ evanescent. Therefore, such spatial frequencies decay rapidly, on a distance scale of several wavelengths, and the observation of sub-wavelength features is essentially impossible using conventional imaging methods.
Over the years, there have been many attempts to bypass the λ/2 limit on optical imaging. Many of these attempts focused on measurements at a very close proximity (“near field”) to the sub-wavelength specimen. One such approach is the Near-field Scanning Optical Microscope (NSOM or SNOM). This technique is based on a very narrow tip, which samples the electromagnetic field point by point at the near field of the sub-wavelength specimen. However, this technique always requires scanning the sample point-by-point, at very high precision (nanometers) and at a very short distance from the sample (sub-microns). Hence, NSOM cannot capture a full image in real time. Other known approaches are based on probing the information with sub-wavelength holes made from thin film of plasmonic metals, and scanning the sample, or using specific arrangements of nano-hole arrays in plasmonic metals to construct super-oscillatory wavepackets in the form of sub-wavelength hot-spots, and then scan the sample at sub-wavelength resolution [1]. Both of these methods rely on scanning, hence cannot yield real-time imaging either. Other techniques for sub-wavelength imaging rely on distributing smaller-than-wavelength fluorescing items on the object and repeating the experiments multiple times [2,3]. All of these techniques suffer from such disadvantages as long scanning, or imaging time involved in scanning, or a need for repeating the experiments, rendering real-time imaging impractical.
Yet another method involves imaging devices (superlens, hyperlens, etc.) made of negative-index materials [4-7]. However, optical negative index materials suffer from huge losses, and in addition the hyperlens can deal only with one-dimensional information, not with full 2D images. Hence, negative-index materials currently do not offer viable technology for sub-wavelength optical imaging.
There have been attempts to achieve sub-wavelength imaging using algorithmic techniques (processing image data). These techniques rely on the analyticity of an EM field: if an analytic function is known exactly at some finite region, it can be completely recovered and uniquely found by analytic continuation. Several concepts and extrapolation methods based on the analytic theory have been developed. However, these methods are extremely sensitive to noise in the measured data and to the assumptions made on the information to be images (signal to be recovered).
Some examples of resolution enhancement techniques are described in US 2008/0260279 providing a method for iterative derivation of a master image from sampled images of non-identical, at least partially overlapping, regions of a scene. The method includes defining a transformation operator mapping positions within the master image to corresponding positions in the sampled image; a distortion operator simulating a modulation transfer function associated with an imaging sensor from which the sampled image was generated; and a sampling operator for reducing an image from the output resolution to the resolution of the sampled image. For each sampled image the transformation operator, distortion operator and sampling operator are applied to a current master image hypothesis to generate a predicted image A difference image is calculated which has pixel values corresponding to the difference in corresponding pixel values between the sampled image and the predicted image. A back-projection of each of the difference images is performed to generate a correction image for the current master image hypothesis. Finally, the correction images are employed to perform a correction to the current master image hypothesis to generate a new master image hypothesis. The correction to the current master image hypothesis includes combining the correction images by deriving a weighted average of values of corresponding pixels in the correction images. The weight of each pixel in each correction image is calculated as a function of a distance as measured in the sampled image between: a point in the sampled image to which the pixel in the correction image is mapped by the transformation operator, and at least one pixel centroid proximal to that point.
Pulse-shape measurement of a short pulse (optical or electronic) signal is another significant example where resolution enhancement of the measurement systems is of much interest. In optics, short laser pulses with durations in the range of nano second to picosecond and femtosecond time-scales are produced regularly. Ultra-short pulses in the attosecond time-scale have been recently produced. In many systems or applications where short pulses are engaged, it is very important to characterize the shape of the pulse (intensity only or amplitude and phase) at high resolution. There are several devices and techniques to measure the pulse-shape of a short laser pulse. For example, high-speed photodiodes or streak camera in conjunction with oscilloscopes are widely used for direct measurements of the pulse-shape (intensity profile) of laser pulses at nanosecond to picosecond temporal resolution because of their simplicity, robustness, relative insensitiveness to the light properties, small size, and low-cost. Several techniques for measuring the pulse-shape of short pulses make use of nonlinear interaction between the pulse and a another pulse with a known pulse-shape (e.g. cross correlation, or cross-correlation frequency resolved optical gating) or with a time-delayed replica of the measured pulse (e.g. autocorrelation, frequency resolved optical gating (FROG), and SPectral Interferometry for Direct E-field Reconstruction which is termed SPIDER).
In a different area of information processing, the past decades have witnessed major breakthroughs in data compression and advances in sampling techniques. Most notably, a new technique was developed in 2006, with the purpose of reducing the sampling rate of information. The technique is called compressed sensing (CS) and it is now widely used for sub-Nyquist sampling of data, and recovering data from a small number of samplings. In doing that, the technique mostly relates to interpolation of information from sub-sampled data, and relies on a single requirement for prior information that the signal (to be recovered) is sparse in a known basis [8-10]. Currently, there are two main mindsets in the field of CS, both mindsets trying to reconstruct a function by a few measurements. The first approach in the CS tries to reconstruct a sparse function by measuring randomly in the Fourier domain of the function. These randomly distributed measurements are aimed at reconstructing the entire function, provided the function is sparse. The second CS approach is aimed at enhancing the resolution of a known low-resolution image. This second technique is based on measurements in the same domain as the original image, and requires that information is retrieved from several defined examples in order to calibrate the algorithm for resolution enhancement. This method can, for example, produce a 200×200 pixels image from an original image with resolution of 20×20 pixels, but still cannot add data which was not in the original image.
Some examples of using the CS technique in signal processing are described in the following patent publications:
U.S. Pat. No. 7,646,924 provides a method and apparatus for compressed sensing yields acceptable quality reconstructions of an object from reduced numbers of measurements. A component x of a signal or image is represented as a vector having m entries. Measurements y, comprising a vector with n entries, where n is less than m, are made. An approximate reconstruction of the m-vector x is made from y. Special measurement matrices allow measurements y=Ax+z, where y is the measured m-vector, x the desired n-vector and z an m-vector representing noise. “A” is an n by m matrix, i.e. an array with fewer rows than columns. “A” enables delivery of an approximate reconstruction, x#, of x. An embodiment discloses approximate reconstruction of x from the reduced-dimensionality measurement y. Given y, and the matrix A, x# of x is possible. This embodiment is driven by the goal of promoting the approximate sparsity of x#.
U.S. Pat. No. 7,511,643 describes a method for approximating a plurality of digital signals or images using compressed sensing. In a scheme where a common component xc of said plurality of digital signals or images an innovative component xi of each of said plurality of digital signals each are represented as a vector with m entries, the method comprises the steps of making a measurement yc, where yc comprises a vector with only ni entries, where ni is less than m, making a measurement yi for each of said correlated digital signals, where yi comprises a vector with only ni entries, where ni is less than m, and from each said innovation components yi, producing an approximate reconstruction of each m-vector xi using said common component yc and said innovative component yi.
US 2009/141995 provides a method of compressed sensing imaging includes acquiring a sparse digital image b, said image comprising a plurality of intensities corresponding to an I-dimensional grid of points, initializing points (x(k), y(k)), wherein x(k) is an element of a first expanded image x defined by b=RΦ−1x, wherein R is a Fourier transform matrix, Φ is a wavelet transform matrix, y(k) is a point in ∂(Σi=11∇iΦ−1x(k))2)1/2, ∇i is a forward finite difference operator for a ith coordinate, and k is an iteration counter; calculating a first auxiliary variable s(k) from x(k)−τ1(αΦΣnLn*yn(k)+ΦR*(RΦ−1x(k)−b)), wherein τ1,α are predetermined positive scalar constants, the sum is over all points n in x, and L* is an adjoint of operator L=(∇1, . . . , ∇1); calculating a second auxiliary variable tn(k) from yn(k)+τ2LnΦ−1x(k), wherein τ2 is a predetermined positive scalar constant; updating x(k+1) from sign (s(k))max {0,|s(k)|−τ1β}, wherein β is a predetermined positive scalar constant; and updating yn(k+1) from min {1/τ2,∥tn(k)∥2}tn(k)/∥tn(k)∥2.
US 2010/0001901 provides for method and apparatus for developing radar scene and target profiles based on Compressive Sensing concept. An outgoing radar waveform is transmitted in the direction of a radar target and the radar reflectivity profile is recovered from the received radar wave sequence using a compressible or sparse representation of the radar reflectivity profile in combination with knowledge of the outgoing wave form. In an exemplary embodiment the outgoing waveform is a pseudo noise sequence or a linear FM waveform.
Also, the compressed sensing technique is described in “Image Super-Resolution as Sparse Representation of Raw Image Patches”, Jianchao Yang, John Wright, Thomas Huang, Yi Ma., IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), 2008.
General Description
There is a need in the art in facilitating resolution enhancement of measurements of signals of various types (e.g. optical, electronic), such as to be beyond a so-called “physical resolution” of measurements. Such physical resolution limitation is typically defined by the response function of a measurement unit (sensor) or a measurement technique (e.g. cross-correlation of measured data with a known reference data).
The resolution with which a signal is collected and detected (measured) in a measurement unit is limited by three main parameters. The first parameter is associated with the sampling of a measurement procedure, such as the sampling rate of a detector, such as spatial resolution of an optical sensor (pixel size in a camera) used for capturing an image, or the temporal sampling rate of detector (e.g. oscilloscope), or the retardation step of an interferometer (or spectrometer). The second parameter is the effective response function of the measurement unit, corresponding to either spatial or temporal frequency response. In this connection, it should be understood that in systems which are linear and shift invariant (e.g. utilizing coherent or completely incoherent illumination) such effective response function is actually represented by a frequency response function itself, while for other system (e.g. non-linear and/or shift variant, e.g. utilizing partially incoherent illumination) the effective response function is determined by a relation between the input signal and the output signal (measured by said measurement system). For example, for measurements of an input field in the form of partially-spatially-incoherent light, there is actually no transfer function, because the operation is not linear and not shift-invariant. Thus, in the present application, the term “frequency response function” or “transfer function” or “spectral response function” should be interpreted broadly meaning the effective response function which in some cases is expressed by a relation between the input and output fields/signals/data. The third parameter is the signal to noise ratio of the measurements.
The inventors have found a technique to restore, from the sensed (measured) data, those features of the input information (input signal) that were lost in the sensed data due to the physical resolution limitation, and cannot be retrieved by just normalizing the measured data by the frequency transfer function of the measurement unit. In other words, the invented technique enables to overcome the limitation associated with the highest frequency in the measured data, at which the signal to noise ratio allows reconstruction through de-convolution (division of the measured data by the spectral transfer function of the measurement system).
The response function of the measurement system and the signal to noise ratio of measurements by said system are the main factors that define and limit the resolution at which the signal is measured. Most often, a spectral response function, g(ω), acts as a low pass filter (LPF) with a characteristic cutoff-frequency fc (where in the time-domain fc˜1/tc, tc being the rise-time of the detector). If the input data (signal or field) contains features at frequencies higher than the cutoff-frequency, then the sensor output signal (measured data) deviates from the input data. In this case, de-convolution methods are often used for extracting the input data. De-convolution methods consist of reversing the detector spectral filtering operation, where the latter is convolution of the input signal with the spectral transfer function of the detector (frequency filtering). De-convolution is achieved by re-amplifying the detector output signal (in the spectral domain) by a factor that corresponds to the inverse of the spectral transfer function of the detector (1/|g(ω)|). This amplification factor becomes very large at high-frequency spectral regions (|g(ω)|<<1). Hence, tiny errors in these spectral regions are very unforgiving because they are amplified by a very large factor, i.e. low noise in such frequencies is amplified (as well as the signal) thus reducing signal to noise ratio (SNR). In fact, de-convolution processes cannot recover information from spectral regions in which SNR(ω)<1/|g(ω)|.
Often, the response function of the system contains a genuine cutoff frequency, above which the transfer function is zero (or corresponds to very large attenuation). The amplitudes of frequencies higher than the cutoff frequency are greatly attenuated, such that these high-frequency signals are below the noise level and cannot be extracted (separated from the noise). Such high frequency information is therefore considered as lost. Thus, it is commonly believed that information in these high frequency spectral regions is lost and cannot be recovered. The frequency at which SNR(ω)<1/|g(ω)| actually presents the effective cutoff frequency of the measurement.
However, high resolution of a measured signal requires high frequency features of the input signal to be measured (i.e., high temporal frequencies for a time-varying signal, or high spatial frequencies in the case of an optical image). For example, in the conventional optical microscopy, an image cannot be captured with resolution higher than the diffraction limit of the optical system (i.e. λ/2 in case of free-space propagation). For example, considering free space propagation of an electromagnetic (EM) wave, if the EM wave propagates a distance z from an object plane to a detector much larger than the wavelength λ, then, since the transfer function of the optical system (CTF for coherent illumination or OTF for incoherent illumination) acts as a low-pass filter, all information carried by spatial frequencies larger than 1/λ are lost.
The present invention provides for reconstructing information (an input signal) at a resolution higher than that defined by the highest frequency of the measuring system or by a ratio between the spectral transfer-function and the signal-to-noise ratio [SNR(ω)<1/|g(ω)|].
In this connection, it should be understood that sensing (measuring) data includes: detection of signals by a suitable sensor unit (detector), where the detected signals may be the input field or those resulted from a known interaction between the input data and reference data (e.g., correlation with known signals); and processing of the measured (detected) data. The detection procedure is to be as accurate as possible, using any suitable detector. Such accurate detection may include any de-coding procedure, provided certain predetermined coding (or pre-processing) of the signal occurred during its propagation to the detector, e.g. at the spectral plane. Such coding or pre-processing may be done by software and/or hardware (e.g. using a mask, e.g. a phase mask or grating). The principles of the invention are applicable to any measured data, irrespective of whether the detection procedure includes de-coding or not. In case the coding/decoding is considered, the invented technique deals with the decoded measured data.
The present invention provides a novel processing technique for processing the measured data to recover details contained in the input information (input signal), that were filtered out in the detection stage, due to the response function of the measurement system (which includes a detector unit and possibly also signal collector(s) on the way to the detector). The filtering out results in that the measured amplitudes of those signals or signal components are so small that dividing them by the transfer response function leads to very large errors.
In other words, with the invented technique, the reconstructed/recovered information contains frequencies higher than the effective frequency cutoff of the effective response function of a measurement system. The reconstruction of measured data (optical field, in the context of optical imaging) according to the invention takes advantages of the principles of the known L1 minimization and compressed sensing techniques, in that it deals with the recovery of information (input signals) that is sparse in some known basis. It should be understood that a sparse signal is such that, in some basis it contains mostly zeros and very few elements differing from zero. The knowledge, or data, about the sparsity of the input field may only contain the fact that the signal is sparse in some basis, and the basis in which the signal is sparse might be a priori known or determined during the measurement procedure (e.g. during the reconstruction of the input field). It is also required that there is a known relation between the signal basis (where the information is sparse) and the measurement base, and back. For some systems, this relation might be written as a transformation operator. However, it should be understood that the present invention does not need such relation to be expressed as an operator, but just needs this relation to be known. The invention properly utilizes the measurement related data, namely data about effective response function of the measurement system, together with the above-described sparsity related data. As indicated above, the invention takes into account the effective response function of the measurement system being expressed by relation between the input field (represented in a basis in which it is sparse) and the output field. In this connection, it should be understood that considering the input field is represented in a basis in which it is sparse (e.g. by some kind of initial processing of the measured data using basic transformation), the relation between the sparsity basis and the measurement basis for linear shift-invariant measurement systems might correspond to the effective response function of the measurement system.
The invention allows for resolution enhancement beyond the effective frequency cutoff of the signal collector (physical limitation of the detection system). The present invention is based on the following: among all signals that can be written as a combination of some known basis functions, which yield the measured results after being “smeared” by the known transfer function (CTF or OTF), the sparsest one of the signals is to be found, i.e. the one comprised of the fewest basis functions. The inventors have termed this novel technique as SMARTER (Sparsity Mediated Algorithmic Reconstruction Technique for Enhanced Resolution) microscopy (in the sub-wavelength case) and SMARTER pulse diagnostics for the characterization of optical pulses.
Thus, according to one broad aspect of the present invention, there is provided a method for reconstructing an input field sensed by a measurement system. The method comprises; providing data (prior knowledge) about the sparsity of the input field (that the input signal (information to be recovered) is sparse in a known basis), and data about effective response function of the measurement system; and processing measured data based on said known data. This “prior knowledge” is used for processing the measured data, generated by the measurement unit, to recover the original information (input signal). The processing stage comprises: a determination of a sparse vector as a function of the following: said data on the sparsity of the input field, said data about the effective response function), and the measured data (output of the measurement system); and using the sparse vector for reconstructing the input information.
The technique of the invention is based on the understanding that, out of all the possibilities of extrapolating the spectrum of the measured data (which all correspond to the same measured data), given the prior knowledge described above, the extrapolation yielding the sparsest input signal (information to be recovered) is unique (in the absence of noise), or, in the presence of noise, provides the recovered information that is very close and the closest to the input information [8-12]. The technique provides for better reconstruction (higher resolution) if the signal basis and the measurement basis are the least correlated, or in other words they are “incoherent with one another” (here the term “incoherence” should not be confused with coherence properties of optical fields). It should, however, be understood that such condition as the least correlation between the signal basis and the measurement basis, while being preferable might be optional for the operational principles and results of the technique of the present invention. Instead, having the measured data occupying the majority of the basis functions in the measurement basis would suffice to recover the input signal properly. When the sparsity basis and the measurement basis are the least correlated, the number of necessary samplings (in the measurement basis) is the smallest.
This technique allows for resolution of the reconstructed input field to be well above an effective frequency cutoff of said response function.
Let us consider for example, optical imaging applications, where prior knowledge about the sparsity of the input field and about the effective response function is provided (i.e. the input image contains a small fraction of non-zero pixels in some known or determined bases, and transformation from the near field to the plane where the data is measured is known). Here, the simplest basis is the near field. The least correlated basis with the near field is the far-field (Fourier plane of the information). Hence, in the optical imaging applications, the technique of the present invention might provider better results if the measurements are taken in the far-field. The transformation relating the near field and the far field is simply the Fourier transform multiplied by the effective transform function which is CTF for coherent fields or OTF for spatially-incoherent fields.
In the same context of optical imaging, one can use another measurement basis: the image plane of an optical imaging system. In this case, the transformation occurred during the input field propagation through the optical measurement system includes the following: Fourier transforming the input signal, multiplying the Fourier transform by the CTF (or OTF), and applying a further Fourier transform (with some magnification) to the results of the multiplication. In this case, the measurement basis and the sparsity basis are identical. Nevertheless, the invented technique still works well, provided that the effective response function satisfies the following condition: the measured data occupies a large fraction of the measurement basis, and the transformation between the two bases is known.
Thus, generally, the present invention is applicable to any measurement basis, as long as the measured data occupies a large fraction of the measurement basis, and the transformation between the two bases is known. The current invention works well in all those measurement bases, provided just that the input image is sparse in a known basis.
For optical imaging applications, the measurement unit is configured for optical measurements, thus including an optical system (lenses etc.) and a suitable optical detector (or camera). As indicated above, the effective response function of such measurement unit is defined by a spatial frequency transfer function which is associated with a Coherent Transfer Function (CTF) for coherent illumination case or an Optical Transfer Function (OTF) for incoherent illumination. Using the method of the invention, the reconstructed input information can have resolution above a cutoff of the OTF (or CTF), which naturally defines the smallest resolvable feature (diffraction limited spot) of the optical imaging system. Preferably, the output field corresponds to a far-field image of the input field.
It should be noted that the present invention provides for reconstruction of signals with non-uniform phase, i.e. signals with varying phases, such as signal with positive phase at one point and negative phase at another, and in the most general sense—signals with phase that can vary arbitrarily between 0 and 2π from one point to another. Thus, no further assumptions, such as non-negativity of the signal, are needed. The recovery of signals (information) with non-uniform phase is done, as part of the above-described reconstruction procedure, by further using an iterative method called nonlocal hard thresholding (NLHT). This technique consists of allocating an off-support of the sparse signal in an iterative fashion, by performing a thresholding step that depends on the values of the neighboring locations (in real space). It should be understood and will be described more specifically further below that generally, that in some embodiments the processor utility of the present invention might be preprogrammed to identify whether the measured signal has uniform or non-uniform phase and accordingly selectively apply either a first processing model that does not utility NLHT (but utilize Basis Pursuit (BP)) or a second processing model that does utilize NLHT; or to eliminate the identification step and utilize the second, more general model utilizing the NLHT. In some other embodiments, where the invention is intended to deal with uniform-phase signals (e.g. dealing with reconstruction of input pulse shapes), the processor utility might utilize only the BP model.
According to another broad aspect of the invention, there is provided a system for reconstructing an input signal. The system includes an input utility which is capable of receiving and storing measured data generated by a measurement unit (being supplied directly therefrom or not). The measured data corresponds to an output signal generated by the measurement unit in response to an input signal which is to be reconstructed. The input unit also receives data indicative of the sparsity of the input field, and data indicative of the effective response function of the measurement unit (e.g. the spatial or temporal response function; or a relation between the input signal and the measurement signal). The system includes a data processor utility which is preprogrammed for analyzing and processing the received data. More specifically, the processor determines a sparse vector which is a function of the following: information about sparsity of the input signal, the measured output signal, and the data about the effective response function of the measurement unit; and uses the sparse vector to reconstruct the input signal by base transformation of the sparse vector onto the original base of the input signal.
According to yet another aspect of the invention, there is provided a system for reconstructing an input optical field. The system includes an input utility capable of receiving and storing measured data (generated by an optical measurement unit and supplied directly therefrom or not) corresponding to an output field generated by the optical measurement unit in response to the input optical field. The input utility also receives data indicative of the sparsity of the input optical field, and data indicative of the effective response function of the optical measurement unit. The system includes a data processor utility which is preprogrammed for analyzing and processing the received data to determines a sparse vector as a function of the sparsity data, the measured date, and the effective response function of the optical measurement unit; and uses the sparse vector to reconstruct the input field by base transformation of the sparse vector onto the original base of the input signal.
In a yet further aspect, the invention provides a measurement system comprising: an imaging system defining an input field propagation to an optical detector, and a control unit having a processor utility for processing measured data output from said detector, the processor utility being configured and operable for processing the measured data based on data about sparsity of the input field and data about an effective response function of the imaging system to reconstruct the input field with resolution above an effective cutoff of said effective response function.
In some other embodiments, the invention is used for processing temporal profile of data, the measured data being that produced by an electronic sensing system. In this case, input field to be measured and reconstructed may be a pulse (e.g. optical pulse), especially a short pulse, namely relatively short as compared to the physical limitation (e.g. rise time) of a pulse detector. A general trend in short pulse-shape measurement techniques and devices is that measurements at higher resolution and larger bandwidth require more complicated and costly devices or systems. Consequently, it is of great interest to increase the resolution of a pulse-shape measurement device or system by post recovery algorithms.
The most attractive application of this aspect of the invention is for reconstructing the profile (intensity profile or amplitude and phase profiles), especially useful for ultra short pulses (in picoseconds range and shorter). In these embodiments, the effective response function of the measurement unit corresponds to a temporal frequency response of the electronic sensing system. Examples of measurement systems/detectors with which the invented technique can advantageously be used include the following: oscilloscope, photodiode, streak camera, cross correlation with a reference filed/signal, auto correlation of the signal/field with itself, frequency resolved optical gating (FROG), Spectral Interferometer for Direct E-field Reconstruction (SPIDER), Cross-correlation Frequency resolved optical gating (XFROG), GRENOUILLE, TADPOLE.
Thus, the invention, in it's yet another broad aspects, provides a method and system to recover the profile of a short pulse (an electronic pulse, an optical pulse, etc.), from measurements taken by a relatively slow measurement system (slow detector) in the meaning that the rise time of the detector is longer than the pulse duration. In these embodiments, the known (initially provided) data includes data about the sparsity of the input pulse and an effective response function (e.g. temporal frequency response function defining a relation between the measurement basis and the signal basis). The shape of the pulse can be complex and its spectrum can include frequencies at spectral regions beyond the effective cutoff frequency for the measurement. Reconstruction uses the measured output signals, the data about the effective transfer function of the detection system, and the fact that short pulses are inherently sparse (in time).
As indicated above, the measured data may include detection of the input field or its interaction with certain reference signal. For example, an optical short pulse can be cross-correlated with another (reference) pulse and the cross correlation signal is measured. The invention provides for reconstructing the input field (recover the structure (amplitude and phase) of a short input pulse, being e.g. electronic or optical pulse) by processing such measured data as well. Reconstruction uses the output cross correlation signal, the known shape of the reference pulse and the interaction model (i.e. cross correlation) between the two pulses.